Equivalent axes

For example, in a plane triangular molecule such as BF3, each of the twofold symmetry axes lying in the plane can be carried into coincidence with each of the others by rotations of 27r/3 or 2 x 2nl3, which are symmetry operations. Thus all three twofold axes are said to be equivalent to one another. In a square planar AB4 molecule, there are four twofold axes in the molecular plane. Two of them, C2 and C2, lie along BAB axes, and the other two, C and Ci, bisect BAB angles. Such a molecule also contains four symmetry planes, each of which is perpendicular to the molecular plane and intersects it along one of the twofold axes. Now it is easy to see that C2 may be carried into C2 and vice versa, and that C2 may be carried into C2 and vice versa, by rotations about the fourfold axis and by reflections in the symmetry planes mentioned, but there is no way to carry C2 or C into either CJ or Cn or vice versa. Thus C2 and C2 form one set of equivalent axes, and and C form another. Similarly, two of the symmetry planes are equivalent to each other, but not to either of the other two, which are, however, equivalent to each other. 

As other illustrations of equivalence and nonequivalence of symmetry elements, we may note that all three of the symmetry planes in BF3 that are perpendicular to the molecular plane are equivalent, as are the three in NH3, whereas the two planes in H20 are not equivalent. The six twofold axes lying in the plane of the benzene molecule can be divided into two sets of equivalent axes, one set containing those that transect opposite carbon atoms and the other set containing those that bisect opposite edges of the hexagon. 

Compatibility relations between states at points on symmetry axes and states at end points of these axes are independent of the particular choice made from a set of equivalent axes. For example, it would make no difference to the compatibility relations in eqs. (5) and (6) if X were to be chosen on kz or kx instead of on ky as in Figure 16.12(b). But there is another kind of compatibility relation which governs states on symmetry axes that he in a plane and which can only be described in relation to a particular choice of coordinate axes. 

Because of the two equivalent axes in dUpsoids of revolution, their hydrodynamics can be described witii only two diffusion coeffidents (Figure 12.2)- Rotation about the uiuque axis in eitiier a prolate or an oblate... 

Taking into account that a crystallographic face can have seven positions in relation to the crystallographic axes (see Figure 2.37) there results at most seven simple forms can be found for the systems with non-equivalent axes (0 7 ) on a given combination (thus, for a certain formula of S5munetry). 

Moreover, an intermediate situation between the hyper-symmetry of the independent molecules and the symmetry of the complete equivalent axes in a crystal generates a case that will be called as pseudo-symmetry. 

The crystal structure of hydroxyapatite, Ca Q(P04)g(0H)2, has been studied by Beevers and McIntyre (1956), Kay et al. (1992), Elliott (1994), and Young et al. (1966). Hydroxyapatite has the hexagonal PG Im space group. This designation refers to the sixfold c-axis perpendicular to three equivalent -axes ( j, 3) at angles of 120° to each other. Ten Ca " ions are located... 

Since the cubic blue phases have three equivalent axes, an applied field breaks the cubic symmetry and creates a preferred axis. But, like the helical phase, blue phases are chiral, being composed of a lattice of double-twist tubes. It is therefore not surprising that applied fields lead to distortion of the lattice (electrostriction) and, for high enough fields, new lower-symmetry phases. These effects occur for both < 0 and a > 0. 

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